Combinatorial Dominance Analysis

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Combinatorial Dominance Analysis
by Yochai Twitto

• Keywords
• Combinatorial Optimization (CO)
• Approximation Algorithms (AA)
• Approximation Ratio (a.r)
• Combinatorial Dominance (CD)
• Domination number/ratio (domn, domr)
• DOM-good approximation
• DOM-easy problem

2
Overview

• Background
• On approximations and approximation ratio.
• Combinatorial Dominance
• What is it ?
• Definitions Notations.
• Problem maximum Cut
• Summary

3
Overview

• Background
• On approximations and approximation ratio.
• Combinatorial Dominance
• What is it ?
• Definitions Notations.
• Problem maximum Cut
• Summary

4
Background

• NP complexity class.
• AA and quality of approximations.
• The classical approximation ratio analysis.
• Example Approximation for TSP.

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NP

• If P ? NP, then finding the optimum of NP-hard
  problem is difficult.

If P NP, P would encompass the NP and
NP-Complete areas.
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Approximations

• So we are satisfied with an approximate solution.
• Question
• How can we measure the solution quality ?

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Solution Quality

• Most of the time, naturally derived from the
  problem definition.
• If not, it should be given as external
  information.

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The classical Approximation Ratio

• (For maximization problem)
• Assume 0 ß 1.
• A.r. ß if
• the solution quality is greater than ßOPT

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ExampleThe Traveling Salesman Problem

• Given a weighted complete graph G, find the
  optimal tour.
• We will assume the graph is metric.
• We will see
• The MST approximation.
• MST approximation ratio analysis.

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MST Approximation for TSP

• Find a minimum spanning tree for G.
• DFS the tree.
• Make shortcuts.

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MST Approx. ratio analysis

• Observation
• If you remove an edge from a tour then you get a
  spanning tree!
• This means that
• Tour cost more than a minimum spanning tree.

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MST Approx. ratio analysis

• Thus, DFSing the MST is of cost
• No more than twice MST cost.
• I.e. no more than twice OPT.
• After shortcuts we get a tour with cost at most
  twice the optimum
• Since the graph is metric.

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Overview

• Background
• On approximations and approximation ratio.
• Combinatorial Dominance
• What is it ?
• Definitions Notations.
• Problem maximum Cut
• Summary

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Combinatorial Dominance

• What is a combinatorial dominance guarantee ?
• Why do we need such guarantees ?
• Example the min partition problem.
• Definitions and notations.

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What is acombinatorial dominance guarantee ?

• A letter of reference
• She is half as good as I am, but I am the best
  in the world
• she finished first in my class of 75 students
• The former is akin to an approximation ratio.
• The latter to combinatorial dominance guarantee.

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What is acombinatorial dominance guarantee ?
(cont.)

• We saw that MST provides a 2-factor
  approximation.
• We can ask
• Is the returned solution guaranteed to be always
  in the top O(n) best solutions ?

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Why do we need that ?

• Let us take another look at the MST approximation
  for TSP.

All other edges of weight 1e (not shown)
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Why do we need that ?

• The spanning tree here is a star.
• DFS Shortcuts yields

OPT 6 4e 6 MST tour size 10 In general
OPT (n-2)(1e) 2 MST 2(n-2) 2
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Why do we need that ?

• But this is the worst possible tour!
• Such kind of analysis is called blackball
  analysis.

Blackball instance
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Corollary

• The approximation ratio analysis gives us only a
  partial insight of the performance of the
  algorithm.
• Dominance analysis makes the picture fuller.

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Simple example of dominance analysis

• The minimum partition problem.
• Greedy-type algorithm.
• Combinatorial dominance analysis of the algorithm.

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ExampleThe minimum partition problem

• Given is a set of n numbers
• V a1, a2, , an
• Find a bipartition (X,Y ) of the indices such
  that
• is minimal.

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Greedy-type algorithm

• Without loss of generality assume
• a1 a2 an .
• Initiate X , Y .
• For j 1, , n
• Add j to X if ,
• Otherwise add j to Y .

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Combinatorial dominance analysis of the
greedy-type algorithm

• Observation
• Any solution produced by the alg. satisfies
  .
• Assume (X ,Y ) is any solution for min
  partition for a2, a3, , an.
• Now, add a1 to Y if ,
• Otherwise add a1 to X .

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Combinatorial dominance analysis of the
greedy-type algorithm (cont.)

• Obtained solution (X ,Y ).
• (X , Y ) is a solution of the original
  problem.
• We have
• Conclusion
• The solution provided by the algorithm dominates
  at least 2n-1 solutions.

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Definitions Notations

• Domination number domn
• Domination ratio domr
• DOM-good approximation
• DOM-easy problem

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Domination Number domn

• Let P be a CO problem.
• Let A be an approximation for P .
• For an instance I of P, the domination number
  domn(I, A) of A on I is the number of feasible
  solutions of I that are not better than the
  solution found by A.

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domn (example)

• STSP on 5 vertices.
• There exist 12 tours
• If A returns a tour of length 7
• then domn(I, A) 8

4, 5, 5, 6, 7, 9, 9, 11, 11, 12, 14, 14 (tours
lengths)
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Domination Number domn

• Let P be a CO problem.
• Let A be an approximation for P .
• For any size n of P, the domination number
  domn(P, n, A) of an approximation A for P is the
  minimum of domn(I, A) over all instances I of P
  of size n.

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Domination Ratio domr

• Let P be a CO problem.
• Let A be an approximation for P .
• Denote by sol(I ) the number of all feasible
  solutions of I.
• For any size n of P, the domination ratio domn(P,
  n, A) of an approximation A for P is the minimum
  of domn(I, A) / sol(I ) taken over all instances
  I of P of size n.

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DOM-good approximation

• A is a DOM-good approximation algorithm for P, if
• It is a polynomial time complexity alg.
• There exists a polynomial p(n) in the size of P,
  such that
• The domination ratio of A is at least 1/p(n) for
  any size n of P.

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DOM-easy problem

• A CO problem is a DOM-easy problem if it admits a
  DOM-good approximation.
• Problems not having this property are DOM-hard.
• Corollary
• Minimum Partition is DOM-easy.
• Furthermore, p(n) is a constant.

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Overview

• Background
• On approximations and approximation ratio.
• Combinatorial Dominance
• What is it ?
• Definitions Notations.
• Problem Maximum Cut
• Summary

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Maximum Cut

• The problem.
• Simple greedy algorithm.
• Combinatorial dominance of the algorithm.
• Well see
• Maximum Cut is DOM-easy.

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Problem Maximum Cut

• Input weighted complete graph G(V, E, w)
• Find a bipartition (X, Y) of V maximizing the sum
• Denote n V.
• Let W be the sum of weights of all edges.

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Problem Maximum Cut

• Denote the average weight of a cut by
• Notice that .
• Next
• Well see a simple algorithm which produces
  solutions that are always better than .
• Well show it is a DOM-good approximation for
  maxCut.

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Algorithm greedy maxCut

• Algrorithm
• Initiate X , Y
• For each j 1n
• Add vj to X or Y so as to maximize its marginal
  value.
• Theorem
• The above algorithm is a 2-factor approximation
  for maxCut.
• Moreover, it produces a cut of weight at least .

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CD analysis

• We will show that the number of cuts of weight at
  most is at least a polynomial part of all cuts
• Call them bad cuts
• Note that this is a general analysis technique.
• Can be applied to another algs./problems

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CD analysis

• A k-cut is a cut (X, Y) for which X k.
• A fixed edge crosses k-cuts.
• Hence the average weight of a k-cut is

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CD analysis

• Let bk be the number of bad k-cuts.
• i.e. k-cuts of weight less than .
• Then

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CD analysis

• Solving for bk we get

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CD analysis

• Hence the number of bad cuts in G is at least
• (by DeMoivre-Laplace theorem)

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CD analysis

• Thus, G has more than bad cuts.
• Corollary
• Maximum Cut is DOM-easy.

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Overview

• Background
• On approximations and approximation ratio.
• Combinatorial Dominance
• What is it ?
• Definitions Notations.
• Problem maximum Cut
• Summary

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Summary
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Summary
MST tour
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Summary

• Domination number domn
• Domination ratio domr
• DOM-good approximation
• DOM-easy problem

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Summary

• Domn(MST, TSP) 1
• Minimum Partition is DOM-easy.
• Maximum Cut is DOM-easy.
• Clique is DOM-hard unless PNP.

blackball
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Combinatorial Dominance Analysis
The End